/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) CR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) QDP
(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(10) YES


----------------------------------------

(0)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

minFM :: Ord b => FiniteMap b a  ->  Maybe b;
minFM EmptyFM = Nothing;
minFM (Branch key _ _ fm_l _) = case minFM fm_l of { 
Nothing-> Just key; 
Just key1-> Just key1; 
 } ;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(1) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case minFM fm_l of {
Nothing  -> Just key;
Just key1  -> Just key1}
"
is transformed to
"minFM0 key Nothing = Just key;
minFM0 key (Just key1) = Just key1;
"

----------------------------------------

(2)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

minFM :: Ord b => FiniteMap b a  ->  Maybe b;
minFM EmptyFM = Nothing;
minFM (Branch key _ _ fm_l _) = minFM0 key (minFM fm_l);

minFM0 key Nothing = Just key;
minFM0 key (Just key1) = Just key1;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(3) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(4)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

minFM :: Ord b => FiniteMap b a  ->  Maybe b;
minFM EmptyFM = Nothing;
minFM (Branch key vy vz fm_l wu) = minFM0 key (minFM fm_l);

minFM0 key Nothing = Just key;
minFM0 key (Just key1) = Just key1;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

----------------------------------------

(6)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

minFM :: Ord b => FiniteMap b a  ->  Maybe b;
minFM EmptyFM = Nothing;
minFM (Branch key vy vz fm_l wu) = minFM0 key (minFM fm_l);

minFM0 key Nothing = Just key;
minFM0 key (Just key1) = Just key1;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="FiniteMap.minFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.minFM wv3",fontsize=16,color="burlywood",shape="triangle"];15[label="wv3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];3 -> 15[label="",style="solid", color="burlywood", weight=9];
15 -> 4[label="",style="solid", color="burlywood", weight=3];
16[label="wv3/FiniteMap.Branch wv30 wv31 wv32 wv33 wv34",fontsize=10,color="white",style="solid",shape="box"];3 -> 16[label="",style="solid", color="burlywood", weight=9];
16 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="FiniteMap.minFM FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
5[label="FiniteMap.minFM (FiniteMap.Branch wv30 wv31 wv32 wv33 wv34)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="Nothing",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="red", weight=0];
7[label="FiniteMap.minFM0 wv30 (FiniteMap.minFM wv33)",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
9 -> 3[label="",style="dashed", color="red", weight=0];
9[label="FiniteMap.minFM wv33",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="FiniteMap.minFM0 wv30 wv4",fontsize=16,color="burlywood",shape="triangle"];17[label="wv4/Nothing",fontsize=10,color="white",style="solid",shape="box"];8 -> 17[label="",style="solid", color="burlywood", weight=9];
17 -> 11[label="",style="solid", color="burlywood", weight=3];
18[label="wv4/Just wv40",fontsize=10,color="white",style="solid",shape="box"];8 -> 18[label="",style="solid", color="burlywood", weight=9];
18 -> 12[label="",style="solid", color="burlywood", weight=3];
10[label="wv33",fontsize=16,color="green",shape="box"];11[label="FiniteMap.minFM0 wv30 Nothing",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="FiniteMap.minFM0 wv30 (Just wv40)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="Just wv30",fontsize=16,color="green",shape="box"];14[label="Just wv40",fontsize=16,color="green",shape="box"];}

----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_minFM(Branch(wv30, wv31, wv32, wv33, wv34), h) -> new_minFM(wv33, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(9) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_minFM(Branch(wv30, wv31, wv32, wv33, wv34), h) -> new_minFM(wv33, h)
The graph contains the following edges 1 > 1, 2 >= 2


----------------------------------------

(10)
YES